[seqfan] Re: Cute interpretation of A002294

[jean-paul allouche]

Dear Allan

Thank you for having mentioned the Bring radical, which
I was not familiar with! I would be in favor of your suggestion
of adding a comment on this I guess.
About the choice of the choices of operations that are or aren't
allowed, you are quite right. For example, it is worth remembering
that, in characteristic p > 0,--if I remember well-- solvable by radicals
mean either the usual or being a root of X^p - X - alpha (if one wants
to keep the equivalence between solvable by radicals and having a
solvable Galois group).
Many explanations about Galois theory skip the case of nonzero
characteristic...

best wishes
jp

Le 11/08/2023 à 22:07, Allan Wechsler a écrit :
> Dear Jean-Paul,
>
> I may have made a sign error, but in any case having the signs flip on odd
> terms of a generating function just means replacing x by -x. And yes, I
> overlooked that comment.
>
> The Bring radical played a part in the story of the proof of the
> unsolvability of quintics in "closed form". Obviously "closed form" is a
> term of art; some operations are allowed and some aren't, and the choice of
> which operations are allowed is a matter of preference and tradition.. If I
> understand the Wikipedia article correctly, adding the Bring radical to the
> set of allowed operations permits a general quintic formula. What's
> interesting is that Erland Bring discussed this function *before* Ruffini
> suggested, and Abel proved, that there was no quintic formula in
> traditional closed form. (And then, of course, Galois turned on his giant
> searchlight that clarified all of this solvablility/constructability
> stuff.) Because of the historical significance of the Bring radical, I
> thought it would be good to have a comment that mentioned the connection,
> but the one you point out might be adequate.
>
> On Fri, Aug 11, 2023 at 2:49 PM jean-paul allouche <
> jean-paul.allouche at imj-prg.fr> wrote:
>
>> Dear Allan
>>
>> Though I have not looked carefully enough, it seems
>> that this is what the sentence
>>   >>>>>
>>   From Stillwell (1995), p. 62: "Eisenstein's Theorem. If y^5 + y = x,
>> then y has a power series expansion y = x - x^5 + 10*x^9/2^1 - 15 * 14 *
>> x^13/3! + 20 * 19 * 18*x^17/4! - ...."
>>   >>>>>
>> probably means (when I said I did not look carefully,
>> I mean that I am not sure that I have understood
>> where the minus signs comme from, and that I
>> have only checked the (absolute value of the)
>> first coefficients in the power series above.
>>
>> best wishes
>> jean-paul
>>
>>
>> Le 09/08/2023 à 03:29, Allan Wechsler a écrit :
>>> According to the Wikipedia article about "Bring radicals", (plus a tiny
>> bit
>>> of elementary algebra),
>>> the generating function for A002294 is the inverse function of f(x) =
>> x^5 +
>>> x. I can't see this mentioned anywhere in the A002294 entry, but perhaps
>>> I'm being blind, and I would appreciate another pair of eyes
>>> double-checking me before I add it.
>>>
>>> If you adjoin this generating function to the usual toolkit of addition,
>>> subtraction, multiplication, division, and exponentiation, then, pace the
>>> Abel-Ruffini theorem, you *can* construct a quintic formula, with which
>> any
>>> quintic can be solved. (Of course the formula is disastrously
>> complicated.
>>> I have not seen it written out.)
>>>
>>> --
>>> Seqfan Mailing list - http://list.seqfan.eu/
>>
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